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Hindawi Publishing CorporationEURASIP Journal on Wireless Communications and NetworkingVolume 2010, Article ID 690161, 14 pagesdoi:10.1155/2010/690161

Research Article

Best Signal Quality in Cellular Networks: Asymptotic Propertiesand Applications toMobility Management in Small Cell Networks

VanMinh Nguyen,1 Francois Baccelli,2 Laurent Thomas,1 and Chung Shue Chen2

1Network and Networking Department, Bell Labs, Alcatel-Lucent, 91620 Nozay, France2TREC (Research Group on Network Theory and Communications), INRIA-ENS, (Ecole Normale Superieure), 75214 Paris, France

Correspondence should be addressed to Van Minh Nguyen, van [email protected]

Received 15 October 2009; Revised 17 February 2010; Accepted 21 March 2010

Academic Editor: Ozgur Oyman

Copyright © 2010 Van Minh Nguyen et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The quickly increasing data traffic and the user demand for a full coverage of mobile services anywhere and anytime are leadingmobile networking into a future of small cell networks. However, due to the high-density and randomness of small cell networks,there are several technical challenges. In this paper, we investigate two critical issues: best signal quality and mobility management.Under the assumptions that base stations are uniformly distributed in a ring-shaped region and that shadowings are lognormal,independent, and identically distributed, we prove that when the number of sites in the ring tends to infinity, then (i) the maximumsignal strength received at the center of the ring tends in distribution to a Gumbel distribution when properly renormalized, and(ii) it is asymptotically independent of the interference. Using these properties, we derive the distribution of the best signal quality.Furthermore, an optimized random cell scanning scheme is proposed, based on the evaluation of the optimal number of sites tobe scanned for maximizing the user data throughput.

1. Introduction

Mobile cellular networks were initially designed for voiceservice. Nowadays, broadband multimedia services (e.g.,video streaming) and data communications have been intro-duced into mobile wireless networks. These new applicationshave led to increasing traffic demand. To enhance networkcapacity and satisfy user demand of broadband services, itis known that reducing the cell size is one of the mosteffective approaches [1–4] to improve the spatial reuse ofradio resources.

Besides, from the viewpoint of end users, full coverage isparticularly desirable. Although today’s macro- and micro-cellular systems have provided high service coverage, 100%-coverage is not yet reached because operators often havemany constraints when installing large base stations andantennas. This generally results in potential coverage holesand dead zones. A promising architecture to cope with thisproblem is that of small cell networks [4, 5]. A small cell onlyneeds lightweight antennas. It helps to replace bulky roof topbase stations by small boxes set on building facade, on publicfurniture or indoor. Small cells can even be installed by end

users (e.g., femtocells). All these greatly enhance networkcapacity and facilitate network deployment. Pervasive smallcell networks have a great potential. For example, Willcomhas deployed small cell systems in Japan [6], and Vodafonehas recently launched home 3G femtocell networks in the UK[7].

In principle, high-density and randomness are the twobasic characteristics of small cell networks. First, reducingcell size to increase the spatial reuse for supporting densetraffic will induce a large number of cells in the samegeographical area. Secondly, end users can set up small cellsby their own means [2]. This makes small cell locations andcoverage areas more random and unpredictable than tra-ditional mobile cellular networks. The above characteristicshave introduced technical challenges that require new studiesbeyond those for macro- and micro-cellular networks. Themain issues concern spectrum sharing and interferencemitigation, mobility management, capacity analysis, andnetwork self-organization [3, 4]. Among these, the signalquality, for example, in terms of signal-to-interference-plus-noise ratio (SINR), and mobility management are two criticalissues.

2 EURASIP Journal on Wireless Communications and Networking

In this paper, we first conduct a detailed study on theproperties of best signal quality in mobile cellular networks.Here, the best signal quality refers to the maximum SINRreceived from a number of sites. Connecting the mobile tothe best base station is one of the key problems. The best basestation here means the base station from which the mobilereceives the maximum SINR. As the radio propagation expe-riences random phenomena such as fading and shadowing,the best signal quality is a random quantity. Investigatingits stochastic properties is of primary importance for manystudies such as capacity analysis, outage analysis, neighborcell scanning, and base station association. However, to thebest of our knowledge, there is no prior art in this area.

In exploring the properties of best signal quality, we focuson cellular networks in which the propagation attenuationof the radio signal is due to the combination of a distance-dependent path-loss and of lognormal shadowing. Considera ring B of radii Rmin and RB such that 0 < Rmin < RB < ∞.The randomness of site locations is modeled by a uniformdistribution of homogeneous density in B. Using extremevalue theory (c.f., [8, 9]), we prove that the maximumsignal strength received at the center of B from n sites inB converges in distribution to a Gumbel distribution whenproperly renormalized and it is asymptotically independentof the total interference, as n → ∞. The distribution of thebest signal quality can thus be derived.

The second part of this paper focuses on applyingthe above results to mobility support in dense small cellnetworks. Mobility support allows one to maintain servicecontinuity even when users are moving around while keepingefficient use of radio resources. Today’s cellular networkstandards highlight mobile-assisted handover in which themobile measures the pilot signal quality of neighbor cellsand reports the measurement result to the network. If thesignal quality from a neighbor cell is better than that of theserving cell by a handover margin, the network will initiate ahandover to that cell. The neighbor measurement by mobilesis called neighbor cell scanning. Following mobile cellulartechnologies, it is known that small cell networking will alsouse mobile-assisted handover for mobility management.

To conduct cell scanning [10–12], today’s cellular net-works use a neighbor cell list. This list contains informationabout the pilot signal of selected handover candidates andis sent to mobiles. The mobiles then only need to measurethe pilot signal quality of sites included in the neighbor celllist of its serving cell. It is known that the neighbor celllist has a significant impact on the performance of mobilitymanagement, and this has been a concern for many years inpractical operations [13, 14] as well as in scientific research[15–18]. Using neighbor cell list is not effective for thescanning in small cell networks due to the aforementionedcharacteristics of high density and randomness.

The present paper proposes an optimized random cellscanning for small cell networks. This random cell scanningwill simplify the network configuration and operation byavoiding maintaining the conventional neighbor cell listwhile improving user’s quality-of-service (QoS). It is alsoimplementable in wideband technologies such as WiMAXand LTE.

In the following, Section 2 describes the system model.Section 3 derives the asymptotic properties and the dis-tribution of the best signal quality. Section 4 presents theoptimized random cell scanning and numerical results.Finally, Section 5 contains some concluding remarks.

2. SystemModel

The underlying network is composed of cells covered by basestations with omnidirectional antennas. Each base stationis also called a site. The set of sites is denoted by Ω ⊂N. We now construct a model for studying the maximumsignal strength, interference, and the best signal quality, afterspecifying essential parameters of the radio propagation andthe spatial distribution of sites in the network.

As mentioned in the introduction, the location of a smallcell site is often not exactly known even to the operator. Thespatial distribution of sites seen by a mobile station will hencebe treated as completely random [19] and will be modeled byan homogeneous Poisson point process [20] with intensity λ.

In the following, it is assumed that the downlink pilotsignal is sent at constant power at all sites. Let Rmin besome strictly positive real number. For any mobile user, it isassumed that the distance to his closest site is at least Rmin andhence the path loss is in the far field. So, the signal strengthof a site i received by a mobile at a position y ∈ R2 is givenby

Pi(y) = A

(∣∣y − xi∣∣)−βXi, for

∣∣y − xi∣∣ ≥ Rmin, (1)

where xi ∈ R2 is the location of site i, A represents thebase station’s transmission power and the characteristics ofpropagation, β is the path loss exponent (here, we consider2 < β ≤ 4), and the random variables Xi = 10X

dBi /10,

which represent the lognormal shadowing, are definedfrom {XdB

i , i = 1, 2, . . .}, an independent and identicallydistributed (i.i.d.) sequence of Gaussian random variableswith zero mean and standard deviation σdB. Typically, σdB

is approximately 8 dB [21, 22]. Here, we consider that fastfading is averaged out as it varies much faster than thehandover decision process.

Cells sharing a common frequency band interfere oneanother. Each cell is assumed allocated no more than onefrequency band. Denote the set of all the cells sharingfrequency band kth by Ωk, where k = 1, . . . ,K . So Ωk∩Ωk′ =∅ for k /= k′, and

⋃Kk=1 Ωk = Ω. The SINR received at y ∈ R2

from site i ∈ Ωk is expressible as

ζi(y) = Pi

(y)

N0 +∑

j /= i, j∈ΩkPj(y) , for i ∈ Ωk, (2)

where N0 is the thermal noise average power which isassumed constant. For notational simplicity, let A := A/N0.Then ζi(y) is given by

ζi(y) = Pi

(y)

1 +∑

j /= i, j∈ΩkPj(y) , for i ∈ Ωk. (3)

In the following, we will use (3) instead of (2).

EURASIP Journal on Wireless Communications and Networking 3

3. Best Signal Quality

In this section, we derive the distribution of the best signalquality. Given a set of sites S ⊂ Ω, the best signal qualityreceived from S at a position y ∈ R2, denoted by YS(y), isdefined as

YS(y) = max

i∈Sζi(y). (4)

Let us first consider a single-frequency network (i.e., K =1).

Lemma 1. In the cell set S of single-frequency network, the sitewhich provides a mobile the maximum signal strength will alsoprovide this mobile the best signal quality, namely,

YS(y) = MS

(y)

1 + I(y)−MS

(y) , ∀y ∈ R2, (5)

where

MS(y) = max

i∈SPi(y)

(6)

is the maximum signal strength received at y from the cell set S,and

I(y) =

∑

i∈ΩPi(y)

(7)

is the total interference received at y.

Proof. Since ζi(y) = Pi(y)/{1+I(y)−Pi(y)} and Pi(y) < I(y),(5) follows from the fact that no matter which cell i ∈ Ω isconsidered, I(y) is the same and from the fact that x/(c − x)with c constant is an increasing function of x < c.

Let us now consider the case of multiple-frequency net-works. Under the assumption that adjacent-channel interfer-ence is negligible compared to cochannel interference, cellsof different frequency bands do not interfere one another.Thus, for a given network topology T , the SINRs receivedfrom cells of different frequency bands are independent. Inthe context of a random distribution of sites, the SINRsreceived from cells of different frequency bands are thereforeconditionally independent given T . Write cell set S as

S =K⋃

k=1

{Sk : Sk ⊂ Ωk}, (8)

with Sk the subset of S allocated to frequency k. Let

YSk

(y) = max

i∈Skζi(y)

(9)

be the best signal quality received at y from sites whichbelong to Sk. The random variables {YSk (y), k = 1, . . . ,K}are conditionally independent given T . As a result,

P{YS(y) ≤ γ | T } =

K∏

k=1

P{YSk

(y) ≤ γ | T }. (10)

Remark 1. For the coming discussions, we define

IS(y) =

∑

i∈SPi(y), (11)

which is the interference from cells in set S. In the following,for notational simplicity, the location variable y appearing inYS(y), MS(y), IS(y), and I(y) will be omitted in case of noambiguity. We will simply write YS, MS, IS, and I . Note thatIS ≤ I since S ⊂ Ω.

Following Lemma 1, the distribution of YS can bedetermined by the joint distribution of MS and I , which isgiven below.

Corollary 1. The tail distribution of the best signal qualityreceived from cell set S is given by

FYS

(γ) =

∫∞

u=0

∫ ((1+γ)/γ)u−1

v=uf(I ,MS)(v,u)dv du, (12)

where f(I ,MS) is the joint probability density of I andMS.

Proof. By Lemma 1, we have

P{YS ≥ γ

} = P{

MS

(1 + I −MS)≥ γ

}

= P

{

I ≤ 1 + γ

γMS − 1

}

=∫∞

u=0

∫ ((1+γ)/γ)u−1

v=uf(I ,MS)(v,u)dv du.

(13)

In view of Corollary 1, we need to study the propertiesof the maximum signal strength MS as well as the jointdistribution of MS and I . As described in the introduction,in dense small cell networks, there could be a large numberof neighbor cells and a mobile may thus receive from manysites with strong enough signal strength. This justifies the useof extreme value theory within this context.

For some Rmin and RB such that 0 < Rmin < RB < ∞, letB ⊂ R2 be a ring with inner and outer radii Rmin and RB,respectively. In this section, we will establish the followingresults.

(1) The signal strength Pi received at the center ofB belongs to the maximum domain of attraction(MDA) of the Gumbel distribution (c.f., Theorem 1in Section 3.1).

(2) The maximum signal strength and the interferencereceived at the center of B from n sites thereinare asymptotically independent as n → ∞ (c.f.,Corollary 3 in Section 3.1).

(3) The distribution of the best signal quality is derived(c.f., Theorem 2 in Section 3.3).


3.1. Asymptotic Properties. To begin with, some technicaldetails need to be specified. Given a ring B as previouslydefined, we will study metrics (such as e.g., signal strength,interference, etc.) as seen at the center of B for a set S ⊂ Ω ofn sites located in B. We will use the notation Mn, Yn, and Ininstead of MS, YS, and IS, respectively, with

Mn = nmaxi=1,i∈S

Pi, In =n∑

i=1,i∈SPi, Yn = n

maxi=1,i∈S

ζi. (14)

Lemma 2. Assume that 0 < Rmin < RB < ∞, that sites areuniformly distributed in B, and that the shadowing Xi followsa lognormal distribution of parameters (0, σX). Then the cdfof the signal strength Pi received at the center of B from a sitelocated in B is given by

FP(x) = c{a−2/βG1(x)− b−2/βG2(x)

−eνx−2/βG3(x) + eνx−2/βG4(x)}

,(15)

where a = AR−βB , b = AR

−βmin, c = A2/β(R2

B − R2min)−1, ν =

2σ2X/β

2, and Gj , j = 1, . . . , 4, refers to the cdf of a lognormaldistribution of parameters (μj , σX), in which

μ1 = log a, μ3 = μ1 +2σ2

X

β,

μ2 = log b, μ4 = μ2 +2σ2

X

β.

(16)

Proof. See Appendix A.

Under the studied system model, {Pi, i = 1, 2, . . .} areindependent and identically distributed (i.i.d.), and so thecdf FMn and probability density function (pdf) fMn of Mn aredirectly obtained as follows.

Corollary 2. Under the conditions of Lemma 2, the cdf and thepdf ofMn are given, respectively, by

FMn(x) = FnP(x), (17)

fMn(x) = n fP(x)Fn−1P (x), (18)

where FP(x) is given by (15), and fP is the pdf of Pi, fP(x) =dFP(x)/dx.

Since Mn is the maximum of i.i.d. random variables,we can also study its asymptotic properties by extremevalue theory. Fisher and Tippett [9, Theorem 3.2.3] provedthat under appropriate normalization, if the normalizedmaximum of i.i.d. random variables tends in distributionto a nondegenerate distribution H , then H must have oneof the three known forms: Frechet, Weibull, or Gumbeldistribution. In the following, we prove that Pi belongs tothe MDA of a Gumbel distribution. First of all, we establishthe following result that is required to identify the limitingdistribution of Mn.

Lemma 3. Under the conditions of Lemma 2, the signalstrength received at the center of B from a site located in B hasthe following tail equivalent distribution:

FP(x) ∼ κexp

(−(log x − μ2

)2/(2σ2

X

))

(log x − μ2

)2/(2σ2

X

) (19a)

∼ κ2√

2πσXG2(x)log x − μ2

, as x −→ ∞, (19b)

where G2(x) = 1 − G2(x), and κ = (σX/√

2πβ)(R2min/(R

2B −

R2min)).

Proof. See Appendix B.

Equation (19b) shows that the tail distribution of thesignal strength Pi is close to that of G2, although it decreasesmore rapidly. The fact that G2 determines the tail behaviorof FP is in fact reasonable, since G2 is the distribution of thesignal strength received from the closest possible neighboring

site (with b = AR−βmin and σX). The main result is given below.

Theorem 1. Assume that 0 < Rmin < RB < ∞, that sites areuniformly distributed in B, and that shadowings are i.i.d. andfollow a lognormal distribution of parameters (0, σX) with 0 <σX < ∞. Then there exists constants cn > 0 and dn ∈ R suchthat

c−1n (Mn − dn)

d−→ Λ as n −→ ∞, (20)

where Λ is the standard Gumbel distribution:

Λ(x) = exp{−e−x}, x ∈ R, (21)

andd→ represents the convergence in distribution. A possible

choice of cn and dn is

cn = σX(2 logn

)−1/2dn,

dn = exp

⎧⎪⎨

⎪⎩μ2 + σX

⎛

⎝√

2 logn +− log logn + log κ

√2 logn

⎞

⎠

⎫⎪⎬

⎪⎭,

(22)

with κ given by Lemma 3.

Proof. See Appendix C.

By Theorem 1, the signal strength belongs to the MDAof the Gumbel distribution, denoted by MDA(Λ). From [23,24], we have the following corollary of Theorem 1.

Corollary 3. Let σ2P be the variance and μP be the mean of

signal strength Pi. Let In = (In − nμP)/(√nσP). Let Mn =

(Mn − dn)/cn, where cn and dn are given by (22). Under theconditions of Theorem 1,

(Mn, In

)d−→ (Λ,Φ) as n −→ ∞, (23)

where Λ is the Gumbel distribution and Φ the standard Gaus-sian distribution, and where the coordinates are independent.


−2 0 2 4 60

0.2

0.4

0.6

0.8

1

Normalised signal strength (decimal scale)

CD

F

Empirical n = 5Analytical n = 5Empirical n = 10

Analytical n = 10Empirical n = 102

Analytical n = 102

(a) Comparison between analytical and empirical CDFs

−6 −4 −2 0 2 4 6

0

0.2

0.4

0.6

0.8

1

Normalised signal strength (decimal scale)

CD

F

Gumbel

Analytical n = 10

Analytical n = 102

Analytical n = 104

(b) Comparison between analytical and limiting CDFs

Figure 1: CDF of Mn under different n: σdB = 8, β = 3.

0

1

JSD

IV

σ (dB)

n = 25n = 50

n = 75n = 100

5 10 15 20

0.2

0.4

0.6

0.8

(a) Under different σdB: β = 3

0 2 4 6 8β

JSD

IV

n = 25n = 50

n = 75n = 100

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(b) Under different β: σdB = 8

Figure 2: Jensen-Shannon divergence between Mn and Λ.

Proof. Conditions 0 < Rmin and σX < ∞ provide σ2P ≤

var{AR−βminXi} < ∞. Then the result follows by Theorem 1and [23, 24].

Note that the total interference I can be written as I =In + Icn where Icn denotes the complement of In in I . Underthe assumptions that the locations of sites are independentand that shadowings are also independent, In and Icn areindependent. The asymptotic independence between Mn and

In thus induces the asymptotic independence between Mn

and I . This observation is stated in the following corollary.

Corollary 4. Under the conditions of Theorem 1,Mn and I areasymptotically independent as n → ∞.

This asymptotic independence facilitates a wide range ofstudies involving the total interference and the maximumsignal strength. This result will be used in the coming


sub-section to derive the distribution of the best signalquality.

Remark 2. The asymptotic properties given by Theorem 1and Corollaries 3 and 4 hold when the number of sites in abounded area tends to infinity. This corresponds to a networkdensification process in which more sites are deployed ina given geographical area in order to satisfy the need forcapacity, which is precisely the small cell setting.

3.2. Convergence Speed of Asymptotic Limits. Theorem 1 andCorollaries 3 and 4 provide asymptotic properties when n →∞. In practice, n is the number of cells to be scanned, andso it can only take moderate values. Thus, it is importantto evaluate the convergence speed of (20) and (23). Wewill do this based on simulations and will measure thediscrepancy using a symmetrized version of the Kullback-Leibler divergence (the so-called Jensen-Shannon divergence(JSdiv)).

Let us start with some numerical evaluations of the con-vergence of Mn to its limiting distribution. Figure 1(a) showsFMn

for different n and compares to empirical simulationresults. As expected the analytical distributions obtained by(17) of Corollary 2 match with the empirical distributionsfor all n. Figure 1(b) plots the analytical distribution andits limiting distribution, that is, the Gumbel distributionΛ. There is a discrepancy in the negative regime (see thecircled region in Figure 1(b)). It is worth pointing out thatas a maximum of signal strengths, Mn ≥ 0 and thus Mn ≥−dn/cn = −

√2 logn/σX since Mn = (Mn−dn)/cn. This means

that FMn(x) = 0, ∀x ≤ −

√2 logn/σX , whereas Λ(x) > 0,

∀x > −∞. This explains the gap observed for small n. Thisdissimilarity should have limited impact as long as we only

deal with positive values of Mn (resp., Mn ≥ −√

2 logn/σX).

We now study the symmetrized divergence betweenthe analytical and limiting distributions of Mn for somemoderate values of n and under different σdB and β. Theconvergence is best for σdB around 10 dB and β aroundtwo to four. For practical systems, σdB is approx. 8 dB and2 < β ≤ 4. We compute the Jensen-Shannon divergencefor β = 3 and σdB = 8 and plot the results in Figures 2(a)and 2(b), respectively. For these (and other) values (withinthe range given above) of σdB and β, Mn, and Λ have lowdivergence.

Let us now measure the (dis)similarity between theempirical joint distribution, P{Mn ≤ u, In ≤ v}, and theproduct of the empirical marginal distributions, P{Mn ≤u}×P{In ≤ v}. Figure 3 shows an example with n = 50, β =3, and σdB = 8. We see that these two density functions arevery similar. Figure 4 compares these two density functionsfor different values of σdB and β. Within the range definedabove, the divergence between the two distributions is againsmall. Comparing Figure 2 and Figure 4, one can conclude

that even if the convergence of Mnd→ Λ remains slow, Mn

and In quickly become uncorrelated. Thus, the independence

between Mn and In holds for moderate values of n, that is,

f(Mn,In)(u, v) ≈ fMn(u)× fIn(v), (24)

and so the same conclusion holds for the independencebetween Mn and I .

3.3. Distribution of the Best Signal Quality. From the aboveresults, we have the distribution of Mn and the asymptoticindependence between Mn and I . In order to derive thedistribution of the best signal quality, we also need thedistribution of the total interference.

Lemma 4. Assume that shadowings are i.i.d. and follow alognormal distribution of parameters (0, σX), β > 2, andthat sites are distributed according to a Poisson point processwith intensity λ outside the disk of radius Rmin > 0. Let Ibe the interference received at the disk center, and φI be thecharacteristic function of I . Then:

(1)

φI(w) = exp

⎧⎨

⎩−πλα(A|w|)α∫ A|w|/Rβ

min

0

1− φX(sign(w)t

)

tα+1dt

⎫⎬

⎭,

(25)

where α = 2/β, and φX is the characteristic function ofXi;

(2) |φI(w)|p ∈ L for all p = 1, 2, . . ., where L is the spaceof absolutely integrable functions;

(3) If AR−βmin is large, then φI admits the following approxi-

mation:

φI(w) ≈ exp(−δ|w|α

[1− j sign(w) tan

(πα

2

)]), (26)

where δ = πλAα exp((1/2)α2σ2X)Γ(1 − α) cos(πα/2),

with Γ(·) denoting the gamma function.

Proof. See Appendix D.

Theorem 2. Under the assumptions of Lemma 4, let B be thering of inner and outer radii Rmin and RB, respectively. Denotethe best signal quality received at the center of B from n sitesin B by Yn. Assume that 0 < σX < ∞, 0 < Rmin < RB < ∞,

that AR−βmin is large, and that πλ(R2

B − R2min) > n, with high

probability, where n is some positive integer. Then the taildistribution of Yn can be approximated by

FYn

(γ) ≈

∫∞

γ

{

fMn(u)∫∞

0

2πw

e−δwα

sin

(

wu− γ

2γ

)

× cos

(

wu + wu− γ

2γ− δwα tan

πα

2

)

dw

}

du.

(27)

Proof. See Appendix E.

The approximation proposed in Theorem 2 will be usedin Section 4 below. It will be validated by simulation in thecontext considered there.


0

0.5

1

5

0

−5 −5

0

5

fnorm. Mn × fnorm. In

norm. Mn

norm. In

(a)

0

0.5

1

5

0

−5 −5

0

5norm. M

nnorm. In

f(norm. Mn ,norm. In)

(b)

Figure 3: Example of joint densities of Mn and In: n = 50, σdB = 8, β = 3. Here, norm. Mn refers to Mn, while norm. In refers to In.

0 5 1 0 1 5

0

J S D

I V

σ ( d B )

n = 25n = 50

n = 75n = 100

0.1

0.2

0.3

0.4

(a) Under different σdB: β = 3

1 1.5 2 2.5 3 3.5 4

−0.5

β

JSD

IV

n = 25n = 50

n = 75n = 100

0.5

(b) Under different β: σdB = 8

Figure 4: Jensen-Shannon divergence between f(Mn ,In) and fMn× fIn .

4. RandomCell Scanning for Data Services

In this section, the theoretical results developed in Section 3are applied to random cell scanning.

4.1. Random Cell Scanning. Wideband technologies such asWiMAX, WCDMA, and LTE use a predefined set of codesfor the identification of cells at the air interface. For example,114 pseudonoise sequences are used in WiMAX [25], while504 physical cell identifiers are used in LTE [26]. When

the mobile knows the identification code of a cell, it cansynchronize over the air interface and then measure the pilotsignal quality of the cell. Therefore, by using a predefinedset of codes, these wideband technologies can have moreautonomous cell measurement conducted by the mobile.In this paper, this identification code is referred to as cellsynchronization identifier (CSID).

In a dense small cell network where a large number ofcells are deployed in the same geographical area, the mobile


can scan any cell as long as the set of CSIDs used in thenetwork is provided. This capability motivates us to proposerandom cell scanning which is easy to implement and hasonly very few operation requirements. The scheme is detailedbelow.

(1) When a mobile gets admitted to the network, its(first) serving cell provides him/her the whole set ofCSIDs used in the network. The mobile then keepsthis information in its memory.

(2) To find a handover target, the mobile randomlyselects a set of n CSIDs from its memory andconducts the standardized scanning procedure ofthe underlying cellular technology, for example,scanning specified in IEEE 802.16 [25], or neighbormeasurement procedure specified in 3G [27] and LTE[12].

(3) The mobile finally selects the cell with the bestreceived signal quality as the handover target.

In the following, we determine the number of cells to bescanned which maximizes the data throughput.

4.2. Problem Formulation. The optimization problem has totake into account the two contrary effects due to the numberof cells to be scanned. On one hand, the larger the set ofscanned cells, the better the signal quality of the chosen site,and hence the larger the data throughput obtained by themobile. On the other hand, scanning can have a linear costin the number of scanned cells, which is detrimental to thethroughput obtained by the mobile.

Let us quantify this using the tools of the previoussections.

Let W be the average cell bandwidth available per mobileand assume that it is a constant. Under the assumption ofadditive white Gaussian noise, the maximum capacity ξn thatthe mobile can have by selecting the best among n randomlyscanned cells is

ξn =W log(1 + Yn). (28)

Hence

E{ξn} =WE

{log(1 + Yn)

}

=W∫∞

γ=0log(1 + γ

)fYn

(γ)dγ,

(29)

where fYn is the pdf of Yn. By an integration by parts oflog(1 + γ) and fYn(γ)dγ = −dFYn(γ), this becomes

E{ξn} =W

∫∞

γ=0

FYn

(γ)

1 + γdγ. (30)

Note that E{ξn} is the expected throughput from thebest cell. Since Yn is the maximum signal quality of then cells, Yn increases with n and so does ξn. Hence, themobile should scan as many cells as possible. However, on theother hand, if scanning many cells, the mobile will consumemuch time in scanning and thus have less time for data

transmission with the serving cell. A typical situation is thatwhere the scanning time increases proportionally with thenumber of cells scanned and where the data transmissionis suspended. This for instance happens if the underlyingcellular technology uses a compressed mode scanning, likefor example, in IEEE 802.16 [25] and also inter-frequencycell measurements defined by 3GPP [12, 27]. In this mode,scanning intervals, where the mobile temporarily suspendsdata transmission for scanning neighbor cells, are interleavedwith intervals where data transmission with the serving cellis resumed.

Another scenario is that of parallel scanning-transmission:here scanning can be performed in parallel to data transmis-sion so that no transmission gap occurs; this is the case in,for example, intrafrequency cell measurements in WCDMA[27] and LTE [12].

Let T be the average time during which the mobile staysin the tagged cell and receives data from it. Let s be the timeneeded to scan one cell (e.g., in WCDMA, the mobile needss = 25 ms if the cell is in the neighbor cell list and s = 800 msif not [28], whereas in WiMAX, s = 10 ms, i.e., two 5-msframes). Let L(n) be the duration of the suspension of datatransmission due to the scanning of the n cells:

L(n) =⎧⎨

⎩

s× n if compressed mode is used,

0 if parallel scan.-trans. is enabled.(31)

Finally, let E{ξ0} be the average throughput received from theserving cell when no scanning at all is performed (this wouldbe the case if the mobile would pick as serving site one of thesites of set S at random).

The gain of scanning n cells can be quantified by thefollowing metric, that we will call the acceleration

ρn � T · E{ξn}

T · E{ξ0}

+ L(n) · E{ξ0}

= T

T + L(n)× E

{ξn}

E{ξ0} .

(32)

In this definition, T · E{ξn} (resp., T · E{ξ0} + L(n) · E{ξ0})is the expected amount of data transmitted when scanning ncells (resp., doing no scanning at all). We aim at finding thevalue of n that maximizes the acceleration ρn.

It is clear that T/(T + L(n)) = 1 when (i) T → ∞,that is, the mobile stays in and receives data from the taggedcell forever, or (ii) L(n) = 0, that is, parallel scanning-transmission is enabled. In these cases, ρn increases withn and the mobile “should” scan as many cells as possible.However, ρn is often concave and the reward of scanning thendecreases. To characterize this, we introduce a growth factorg defined as follows:

gn � ρnρn−1

= T + L(n− 1)T + L(n)

× E{ξn}

E{ξn−1

} . (33)

Special cases as those considered above can be cast withina general framework which consists in finding the value of nthat maximizes ρn under the constraint that gn ≥ 1 + Δg ,where Δg > 0 is a threshold.


0 50 100 150 200 250

10−1

100

n (number cells scanned)

NumericalSimulation

E{ξ

n}/

E{ξ

250}

(a) Plot of E{ξn}/maxk E{ξk}

100 101 1020

0.2

0.4

0.6

0.8

1


NumericalSimulation

ρ n/m

ax25

0n=1

(ρn

)

nnumerical = 43nSimulation = 42

(b) Plot of ρn/maxk ρk , T = 0.5 second

0 20 40 60 80 1000

20

40

60

80

100

T (s)

Opt

imaln

(c) Optimal number of cells to be scanned

100 101 102

1

1.2

1.4

1.6

1.8

2


g n

Limiting caseT = 5 s

T = 0.5 sT = 0.2 s

(d) Growth factor gn under different T

Figure 5: Numerical results in the random cell scanning optimization.

4.3. Numerical Result. In the following, we show how toapply the above results to find the optimal n. We adoptWCDMA as the underlying cellular network technology. 100omnidirectional small cell base stations are deployed in asquare domain of 1 km × 1 km. The network density is thusequal to

λ = 10−4 small base stations/m2. (34)

It is assumed that any cell synchronization identifier canbe found in a radius RB = 1 km. We take Rmin equal to 2

meters. The propagation path loss is modeled by the picocellpath loss model [29]:

PL[dB](d) = 37 + 30log10(d) + 18.3 f (( f +2)/( f +1)−0.46), (35)

where d is the distance from the base station in meters, fthe number of penetrated floors in the propagation path.For indoor office environments, f = 4 is the default value[22]; however, here, the small cell network is assumed to bedeployed in a general domain including outdoor urban areaswhere there are less penetrated walls and floors. So, we usef = 3 in our numerical study.


It is assumed that the total transmission power includingthe antenna gain of each small cell base station is PTx,[dBm] =32 dBm. Shadowing is modeled as a random variable withlognormal distribution with an underlying Gaussian distri-bution of zero mean and 8 dB standard deviation. The signalstrength received at any distance d from a base station i isexpressible as

Pi,[dBm](d) = PTx,[dBm] − PL[dB](d) + XdBi . (36)

By (35), we have

Pi,[dBm](d) = PTx,[dBm] − 37− 18.3 f (( f +2)/( f +1)−0.46)

− 30log10(d) + XdBi .

(37)

The parameters A and β appearing in Pi = Ad−βXi canbe identified from (37) after converting the received signalstrength from the dBm scale to the linear scale:

β = 3, A[mW] = 10(PTx,[dBm]−37−18.3 f (( f +2)/( f +1)−0.46))/10. (38)

The received noise power N0 is given by

N0,[mW] = kBTKelvin ×NF[W] ×W[Hz] × 103, (39)

where the effective bandwidth W[Hz] = 3.84 × 106 Hz, kB

is the Boltzmann constant, and TKelvin is the temperature inKelvin, kBTKelvin = 1.3804× 10−23 × 290 W/Hz and NF[dB] isequal to 7 dB.

It is assumed that the mobile is capable of scanningeight identified cells within 200 ms [28]. So, the average timeneeded to scan one cell is given by s = 25 ms.

In order to check the accuracy of the approximationsused in the analysis, a simulation was built with the aboveparameter setting. The interference field was generatedaccording to a Poisson point process of intensity λ in aregion between Rmin and R∞ = 100 km. For a number n,the maximum of SINR received from n base stations whichare randomly selected from the disk B between radii Rmin

and RB was computed. After that the expectation of themaximal capacity E{ξn} received from the n selected BSs wasevaluated.

In Figure 5(a), the expectation of the maximal through-put E{ξn} for different n is plotted, as obtained through theanalytical model and simulation. The agreement betweenmodel and simulation is quite evident. As shown inFigure 5(a), E{ξn} increases with n, though the increasingrate slows down as n increases. Note that in Figure 5(a),E{ξn} is plotted after normalization by E{ξ250}.

Figure 5(b) gives an example of acceleration ρn for T =0.5 second and L(n) = n×25 ms. In the plot, ρn is normalizedby its maximum. Here, an agreement between model andsimulation is also obtained. We see that ρn first increasesrapidly with n, attains its maximum at n = 42 by simulationand n = 43 by model, and then decays.

Next, using the model we compute the optimal numberof cells to be scanned and the growth factor gn for different

T . Note that in (32), the factor T/(T +L(n)) can be rewrittenas

T

T + L(n)= 1

1 + n× s/Tfor

⎧⎨

⎩

T <∞,

L(n) = n× s.(40)

It is clear that this factor also depends on the ratio T/s.Figure 5(c) plots the optimal n for different values of T/s.Larger T/s will drive the optimal n towards larger values.Since T can be roughly estimated as the mobile residencetime in a cell, which is proportional to the cell diameterdivided by the user speed, this can be rephrased by statingthat the faster the mobile, the smaller T and thus the fewercells the mobile should scan.

Finally, Figure 5(d) plots the growth factor gn withdifferent T . In Figure 5(d), the “limiting case” correspondsto the case when T → ∞ or L(n) = 0. We see that gn isquite stable w.r.t. the variation of T . Besides, gn flattens outat about 30 cells for a wide range of T . Therefore, in practicethis value can be taken as a recommended number of cells tobe scanned in the system.

5. Concluding Remarks

In this paper, we firstly develop asymptotic properties of thesignal strength in cellular networks. We have shown that thesignal strength received at the center of a ring shaped domainB from a base station located in B belongs to the maximumdomain of attraction of a Gumbel distribution. Moreover,the maximum signal strength and the interference receivedfrom n cells in B are asymptotically independent as n →∞. The above properties are proved under the assumptionthat sites are uniformly distributed in B and that shadowingis lognormal. Secondly, the distribution of the best signalquality is derived. These results are then used to optimizescanning in small cell networks. We determine the number ofcells to be scanned for maximizing the mean user throughputwithin this setting.

Appendices

A. Proof of Lemma 2

Let di = |y − xi| be the distance from a site located atxi ∈ R2 to a position y ∈ R2. Under the assumption thatsite locations are uniformly distributed in B, the distance difrom a site located in B, that is, xi ∈ B, to the center of B hasthe following distribution:

FD(d) = P[di ≤ d] = πd2 − πR2min

πR2B − πR2

min= d2 − R2

min

R2B − R2

min. (A.1)

Let Ui = Ad−βi , for β > 0, its distribution is equal to

FU(u) = −c(u−2/β − a−2/β

), for u ∈ [a, b], (A.2)

where c = A2/β(R2B −R2

min)−1, a = AR−βB , and b = AR

−βmin. The

density of Ui is given by fU(u) = (2c/β)u−1−2/β.


Thus, the distribution FP of the power Pi is equal to

FP(x) =∫ b

u=aFX

(x

u

)fU(u)du. (A.3)

Substituting FX with lognormal distribution of parame-ters (0, σX) and fU given above into (A.3), after changing thevariable such that t = log(x/u), we have

FP(x) = c

βx−2/β

(∫ log(x/a)

log(x/b)e2t/βdt

+∫ log(x/a)

log(x/b)e2t/β erf

(t√2σX

)dt

)

,

(A.4)

where the first integral is straightforward. By doing anintegration by parts of erf(t/

√2σX) and e2t/βdt for the second

integral, we get

FP(x) = c

βx−2/β β

2

[

e2t/β + e2t/β erf(

t√2σX

)

−e2σ2X /β

2erf

(t√2σX

−√

2σXβ

)]∣∣∣∣∣

log(x/a)

t=log(x/b)

.

(A.5)

After some elementary simplifications, we can obtain

FP(x) = c{a−2/β

(12

+12

erf(

log x − μ1√2σX

))

− b−2/β(

12

+12

erf(


))

+ eνx−2/β[−1

2− 1

2erf(


)

+12

+12

erf(


)]},

(A.6)

where ν = 2σ2X/β

2, μ1 = log a, μ3 = μ1 + 2σ2X/β, μ2 = log b,

and μ4 = μ2 + 2σ2X/β. Let Gj , j = 1, . . . , 4, be the lognormal

distribution of parameters (μj , σX), j = 1, . . . , 4, FP can berewritten as (15).

B. Proof of Lemma 3

Let Gj(x) = 1−Gj(x) and note that c(a−2/β − b−2/β) = 1, wehave from (15)

FP(x) = 1− c{a−2/βG1(x)− b−2/βG2(x)

−eνx−2/βG3(x) + eνx−2/βG4(x)}.

(B.1)

This yields the tail distribution Fp = 1− FP :

Fp(x) = c{a−2/βG1(x)− b−2/βG2(x)

−eνx−2/βG3(x) + eνx−2/βG4(x)}.

(B.2)

For (B.2), we have Gj(x) = (1/2) erfc{(log x −μj)/(

√2σX)}. An asymptotic expansion of erfc(x) for large

x [30, 7.1.23] gives us:

G3(x) ∼∞σX√

2π(log x − μ3

) exp

{

−(log x − μ3

)2

2σ2X

}

= σX√2π(log x − μ1 − 2σ2

X/β)

× exp

⎧⎨

⎩−(log x − μ1 − 2σ2

X/β)2

2σ2X

⎫⎬

⎭

= σXa−2/βe−νx2/β√

2π(log x − μ1

) exp

{

−(log x − μ1

)2

2σ2X

}

× 11− 2σ2

X/β(log x − μ1

) ,

(B.3)

in which after a Taylor expansion of the last term on theright-hand side, we can have

eνx−2/βG3(x)

∼∞ a−2/β

⎡

⎣G1(x) +

√2π

σ3X

β

exp(−(log x − μ1

)2/2σ2

X

)

(log x − μ1

)2

⎤

⎦.

(B.4)

This implies that

a−2/βG1(x)− eνx−2/βG3(x)

∼∞ −√

2π

a−2/βσ3X

β


)2/2σ2

X

)

(log x − μ1

)2 .(B.5)

In the same manner, we have

b−2/βG2(x)− eνx−2/βG4(x)

∼∞ −√

2π

b−2/βσ3X

β


)2/2σ2

X

)

(log x − μ2

)2 .(B.6)

A substitution of (B.5) and (B.6) into (B.2) results in

Fp(x) ∼∞

√2π

cσ3X

β

⎧⎨

⎩b−2/β


)2/(2σ2

X

))

(log x − μ2

)2

−a−2/βexp

(−(log x − μ1

)2/(2σ2

X

))

(log x − μ1

)2

⎫⎬

⎭.

(B.7)

Moreover, b > a yields μ2 − μ1 = log(b/a) > 0. Then, wehave the following result for large x:


)2/2σ2

X

)/(log x − μ1

)2


)2/2σ2

X

)/(log x − μ2

)2

=(

log x − μ2

log x − μ1

)2

exp

(μ2

2 − μ21

2σ2X

)

x−(μ2−μ1)/σ2X −→

x→∞ 0.

(B.8)


Taking this into account in (B.7), finally we have

Fp(x) ∼∞ κexp

(−(log x − μ2

)2/(2σ2

X

))

((log x − μ2)/(

√2σX)

)2

∼∞ 2√

2πσXκG2(x)

log x − μ2, κ = σX√

2πβR2

min

R2B − R2

min.

(B.9)

C. Proof of Theorem 1

We will use Lemma 3 and the following two lemmas to proveTheorem 1.

Lemma 5 (Embrechts et al. [9]). Let Zi be i.i.d. randomvariables having distribution F, and Ψn = maxni=1Zi. Let gbe an increasing real function, denote Zi = g(Zi), and Ψn =maxni=1Zi. If F ∈ MDA(Λ) with normalizing constant cn anddn, then

limn→∞P

(Ψn ≤ g(cnz + dn)

)= Λ(z), z ∈ R. (C.1)

Lemma 6 (Takahashi [31]). Let F be a distribution function.Suppose that there exists constants ω > 0, l > 0, η > 0, andr ∈ R such that

limx→∞

(1− F(x))(lxre−ηxω)

= 1. (C.2)

For μ ∈ R and σ > 0, let F∗ = F((x − μ)/σ). Then,F∗ ∈ MDA(Λ) with normalizing constants c∗n = σcn andd∗n = σdn + μ, where

cn =(logn/η

)1/ω−1

ωη,

dn =(

lognη

)1/ω

+η1/ω

ω2

r(log logn− logη

)+ ω log l

(logn

)1−1/ω .

(C.3)

Let g(t) = e√

2σX t+μ2 be a real function defined on R, gis increasing with t. Let Qi be the random variable such thatPi = g(Qi). By (19a) of Lemma 3, the tail distribution FQ isgiven by

FQ(x) = FP

(e√

2σXx+μ2

)∼ κx−2e−x

2, as x −→ ∞. (C.4)

By (C.4), FQ satisfies Lemma 6 with constants l = κ, r =−2, η = 1, and ω = 2. So, FQ ∈ MDA(Λ) with the followingnormalizing constants:

c∗n =(logn/η

)1/ω−1

ωη= 1

2

(logn

)−1/2,

d∗n =(

lognη

)1/ω

+η1/ω

ω2

r(log logn− logη

)+ ω log l

(logn

)1−1/ω

= (logn)1/2 +

12

(− log logn + log κ)

(logn

)1/2 .

(C.5)

Then, by Lemma 5, we have

limn→∞P

(Mn ≤ g

(c∗n x + d∗n

)) = Λ(x), x ∈ R. (C.6)

By a Taylor expansion of exp(√

2σXc∗n x), we have

limn→∞P

{e−(

√2σXd∗n +μ2)Mn ≤ 1 +

√2σXc∗n x + o

(c∗n)} = Λ(x).

(C.7)

Since c∗n → 0 when n → ∞, we have

Mn − e√

2σXd∗n +μ2

√2σXc∗n e

√2σXd∗n +μ2

d−→ Λ, as n −→ ∞. (C.8)

Substituting c∗n and d∗n from (C.5) into (C.8), we obtaincn and dn for (22). The conditions Rmax < ∞, Rmin > 0 andσX > 0 provide κ > 0. This leads to dn > 0, and consequently,cn > 0.

D. Proof of Lemma 4

Under the assumptions of the lemma, the interference fieldcan be modeled as a shot noise defined on R2 excluding theinner disk of radius Rmin. Hence, using Proposition 2.2.4 in[20], the Laplace transform of I is given by

LI(s) = exp

{

−2πλ∫∞

Rmin

(1− E

{e−sAXi/rβ

})r dr

}

. (D.1)

Noting that

φI(w) = LI(− jw

), w ∈ R, (D.2)

we have from (D.1) that

φI(w) = exp

{

−2πλ∫∞

Rmin

(1− E

{e jwAXi/rβ

})r dr

}

. (D.3)

Using the change of variable t = |w|Ar−β, we obtain∫ +∞

Rmin

(1− E

{e jwAXi/rβ

})r dr

= (A|w|)2/β

β

∫ A|w|/Rβmin

0

1− E{e j sign(w)tXi

}

t2/β+1 dt,

(D.4)

where E{e j sign(w)tXi} = φX(sign(w)t). So, substituting thisinto (D.3), we get the first part of the Lemma 4.

From (25), for all p = 1, 2, . . ., we have∣∣φI(w)

∣∣p

= exp

⎛

⎝−pπλα(A|w|)αE⎧⎨

⎩

∫ A|w|/Rβmin

0

1− cos(tXi)tα+1

dt

⎫⎬

⎭

⎞

⎠.

(D.5)

Since 1− cos(tXi) ≥ 0, ∀t ∈ R, we have

E

⎧⎨

⎩

∫ A|w|/Rβmin

0

1− cos(tXi)tα+1

dt

⎫⎬

⎭ ≥ 0. (D.6)


Therefore∣∣φI(w)

∣∣p ≤ exp(−c|w|α), (D.7)

where c is some positive constant, and hence the right hand-side of this is an absolutely integrable function. This provesthe second assertion of Lemma 4.

Under the assumption that AR−βmin ≈ ∞, φI can be

approximated by

φI(w) ≈ exp

(

−πλα(A|w|)α∫∞

0

1− φX(sign(w)t

)

tα+1dt

)

.

(D.8)

For 0 < α < 1, we have∫∞

0

1− e j sign(w)tXi

tα+1dt

= −Γ(−α)∣∣sign(w)Xi

∣∣αe− j sign(w)(πα/2).

(D.9)

Since Xi ≥ 0, we can write | sign(w)Xi|α = Xαi . Taking

expectations on both sides, we get

∫ +∞

0

1− E{e j sign(w)tXi

}

tα+1dt

= −E{Xαi

}Γ(−α)e− j sign(w)(πα/2)

= E{Xαi

}Γ(1− α)α

cos(πα

2

)[1− j sign(w) tan

(πα

2

)].

(D.10)

Substituting this into (D.8) and noting that

E{Xαi

} = exp(

12α2σ2

X

), (D.11)

for Xi lognormally distributed, we obtain (26).

E. Proof of Theorem 2

Under the assumption that sites are distributed as a homoge-neous Poisson point process of intensity λ in B, the expectednumber of cells in B is πλ(R2

B − R2min). We assume that

πλ(R2B − R2

min) is much larger than n, which ensures thatthere are n cells in B with high probability, so that Yn is welldefined.

Under the conditions of Theorem 2, Mn and I areasymptotically independent according to Corollary 4. So, bysubstituting (24) into (12), we have

P{Yn ≥ γ

} ≈∫∞

0fMn(u)

∫∞

0h(v,u) fI(v)dv du, (E.1)

where

h(v,u) = 1(v≤(1+γ)u/γ−1)1(v≥u)

=

⎧⎪⎪⎨

⎪⎪⎩

1 if v ∈[

u,1 + γ

γu− 1

]

, u > γ

0 otherwise.

(E.2)

It is easily seen that h(v,u) is square-integrable withrespect to v, and its Fourier transform w.r.t. v is given by

h(w

2π,u)=

⎧⎪⎪⎨

⎪⎪⎩

0 if u ≤ γ,∫ ((1+γ)/γ)u−1

ue− jwvdv if u > γ,

(E.3)

which yields:

h(w

2π,u)=

⎧⎪⎪⎨

⎪⎪⎩

0 if u ≤ γ,

1jw

(e− jwu − e jw(1−((1+γ)/γ)u)

)if u > γ.

(E.4)

Besides, according to Lemma 4 we have that φI ∈ L andφI ∈ L2, where L2 is the space of square integrable functions.And so, by Theorem 3 in [32, page 509], fI is boundedcontinuous and square integrable. Hence, by applying thePlancherel-Parseval theorem to the inner integral of (E.1),we have

∫∞

0h(v,u) fI(v)dv =

∫∞

−∞h(−w,u) fI(w)dw, (E.5)

where fI(w) is the Fourier transform of fI(v). Take (E.4) intoaccount for (E.5) and (E.1), we have

FYn

(γ) =

∫∞

γ

{fMn(u)

∫∞

−∞h(−w,u) fI(w)dw

}du, (E.6)

where we further have∫∞

−∞h(−w,u) fI(w)dw

= 12π

∫ +∞

−∞h(− w

2π,u)fI

(w

2π

)dw

= 12π

∫ +∞

0

(h(w

2π,u)fI

(−w2π

)+ h(−w

2π,u)fI

(w

2π

))dw.

(E.7)

Note that

fI

(w

2π

)= φI(−w). (E.8)

And under the assumption that AR−βmin ≈ ∞, φI is approx-

imated by (26). Thus, by (26) and (E.4), we have for w ∈[0, +∞):

h(w

2π,u)fI

(− w

2π

)

≈ e−δwα

jw

{e j(−wu+δwα tan(πα/2))

− e− j(−w+w((1+γ)/γ)u−δwα tan(πα/2))}

,

h(− w

2π,u)fI

(w

2π

)

≈ e−δwα

jw

{−e− j(−wu+δwα tan(πα/2))

+ e j(−w+w((1+γ)/γ)u−δwα tan(πα/2))}.

(E.9)


By (E.9), we get

12π

[h(w

2π,u)fI

(−w2π

)+ h(−w

2π,u)fI

(w

2π

)]

≈ e−δwα

πw

[sin(−wu + δwα tan

πα

2

)

+ sin

(

−w + w1 + γ

γu− δwα tan

πα

2

)]

= 2e−δwα

πwsin

(

wu− γ

2γ

)

× cos

(

wu + wu− γ

2γ− δwα tan

πα

2

)

.

(E.10)

Substitute the above into (E.7) and then into (E.6), we have(27).

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